A subgroup H of a finite group G is called semipermutable if it is permutable with every subgroup K of G with （｜H｜ ｜K｜） = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with （p, ｜...A subgroup H of a finite group G is called semipermutable if it is permutable with every subgroup K of G with （｜H｜ ｜K｜） = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. In this paper, we investigate the influence of s-semipermutablity of some subgroups of prime power order of a finite group on its supersolvablility.展开更多
A subgroup H of a finite group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this paper, we investigate the structure ...A subgroup H of a finite group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this paper, we investigate the structure of a group under the assumption that every subgroup with order pm of a Sylow p-subgroup P of G is SS-quasinormal in G for a fixed positive integer m. Some interesting results related to the p-nilpotency and supersolvability of a finite group are obtained. For example, we prove that G is p-nilpotent if there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| or 2|D| whenever p = 2 and |D| = 2 is SS-quasinormal in G, where p is the smallest prime dividing the order of G and P is a Sylow p-subgroup of G.展开更多
Let H be a subgroup of a group G. Then H is said to be S-quasinormal in G if HP = PH for every Sylow subgroup P of G; H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup...Let H be a subgroup of a group G. Then H is said to be S-quasinormal in G if HP = PH for every Sylow subgroup P of G; H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing the order of H. In this paper, we say that H is weakly S-embedded in G if G has a normal subgroup T such that HT is an S-quasinormal subgroup of G and H VIT ≤ HSE, where HSE denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. Some results about the influence of weakly S-embedded subgroups on the structure of finite groups are given.展开更多
In this note,we show that if every maximal subgroup of a Sylow p-subgroup of a finite group has a p-solvable supplement then the group is necessarily p-solvable.This gives a positive answer to Problem 17.111 of the Ko...In this note,we show that if every maximal subgroup of a Sylow p-subgroup of a finite group has a p-solvable supplement then the group is necessarily p-solvable.This gives a positive answer to Problem 17.111 of the Kourovka Notebook(Unsolved Problems in Group Theory),which was posed by Skiba.展开更多
基金NNSF of China (10471085)NSF of Shanxi Province of China (20011004)Key Proj. of Ministry of Education(02023)the Returned Overseas Students Foundation of Shanxi Province of China ([2004]7)
文摘A subgroup H of a finite group G is called semipermutable if it is permutable with every subgroup K of G with （｜H｜ ｜K｜） = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. In this paper, we investigate the influence of s-semipermutablity of some subgroups of prime power order of a finite group on its supersolvablility.
基金supported by National Natural Science Foundation of China (Grant No. 10771132)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200802800011)+1 种基金the Research Grant of Shanghai University, Shanghai Leading Academic Discipline Project (Grant No. J50101)Natural Science Foundation of Anhui Province (Grant No.KJ2008A030)
文摘A subgroup H of a finite group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this paper, we investigate the structure of a group under the assumption that every subgroup with order pm of a Sylow p-subgroup P of G is SS-quasinormal in G for a fixed positive integer m. Some interesting results related to the p-nilpotency and supersolvability of a finite group are obtained. For example, we prove that G is p-nilpotent if there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| or 2|D| whenever p = 2 and |D| = 2 is SS-quasinormal in G, where p is the smallest prime dividing the order of G and P is a Sylow p-subgroup of G.
基金supported by National Natural Science Foundation of China (Grant Nos.10771172,11001226)Postgraduate Innovation Foundation of Southwest University (Grant Nos. ky2009013,ky2010007)
文摘Let H be a subgroup of a group G. Then H is said to be S-quasinormal in G if HP = PH for every Sylow subgroup P of G; H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing the order of H. In this paper, we say that H is weakly S-embedded in G if G has a normal subgroup T such that HT is an S-quasinormal subgroup of G and H VIT ≤ HSE, where HSE denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. Some results about the influence of weakly S-embedded subgroups on the structure of finite groups are given.
基金supported by National Natural Science Foundation of China(Grant Nos.11101055 and 11171364)
文摘In this note,we show that if every maximal subgroup of a Sylow p-subgroup of a finite group has a p-solvable supplement then the group is necessarily p-solvable.This gives a positive answer to Problem 17.111 of the Kourovka Notebook(Unsolved Problems in Group Theory),which was posed by Skiba.
